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Related Rates01:18

Related Rates

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When two or more physical quantities are linked by a single relationship, a change in one variable necessarily affects the others. This interdependence forms the basis of related rates analysis, which examines how different quantities change with respect to time. A classic physical example is an expanding balloon, where the size of the balloon changes continuously as air is added.For a hot air balloon, the inflated envelope is commonly idealized as a perfect sphere to simplify mathematical...
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False memories represent a cognitive distortion in which individuals recall events that did not happen, or remember them in an altered form. This phenomenon highlights the brain's constructive nature in processing and recalling memories, emphasizing that memory is not a perfect representation of past events but rather a dynamic reconstruction influenced by various factors.
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Drug discovery is a multifaceted process involving extensive screening, testing, and optimization of lead compounds to identify potential new drugs for therapeutic use. It combines several approaches, including screening large numbers of natural products, chemical modification of known active molecules, identification of new drug targets, and rational design based on biological mechanisms and drug-receptor structure. These approaches are carried out in both academic research laboratories and...
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The rate of reaction is the change in the amount of a reactant or product per unit time. Reaction rates are therefore determined by measuring the time dependence of some property that can be related to reactant or product amounts. Rates of reactions that consume or produce gaseous substances, for example, are conveniently determined by measuring changes in volume or pressure.
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Polarimetry finds application in chemical kinetics to measure the concentration and reaction kinetics of optically active substances during a chemical reaction. Optically active substances have the capability of rotating the plane of polarization of linearly polarized light passing through them—a feature called optical rotation. Optical activity is attributed to the molecular structure of substances. Normal monochromatic light is unpolarized and possesses oscillations of the electrical...
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False discovery rates: a new deal.

Matthew Stephens

    Biostatistics (Oxford, England)
    |October 21, 2016
    PubMed
    Summary

    This study introduces a novel Empirical Bayes method for large-scale hypothesis testing and false discovery rate (FDR) estimation. The approach enhances accuracy by assuming unimodal effects and using both effect size and standard error for more robust statistical inference.

    Area of Science:

    • Statistical inference
    • Bioinformatics
    • Genomics

    Background:

    • Large-scale hypothesis testing is crucial in modern data analysis.
    • Existing methods for False Discovery Rate (FDR) estimation often use limited input data.
    • Incorporating prior knowledge about effect size distributions can improve statistical power.

    Purpose of the Study:

    • To develop a new Empirical Bayes approach for large-scale hypothesis testing and FDR estimation.
    • To improve the accuracy and robustness of statistical inference in high-dimensional data.
    • To provide interval estimates for effect sizes and introduce a new significance measure, the local false sign rate.

    Main Methods:

    • An Empirical Bayes framework is proposed, incorporating a unimodal assumption for effect size distributions.
    Keywords:
    Empirical BayesFalse discovery ratesMultiple testingShrinkageUnimodal

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  • The method utilizes both effect size estimates and standard errors as input for each test.
  • Convex optimization is employed for efficient estimation of the unimodal distribution.
  • The local false sign rate is introduced as a novel measure of significance.

    • Main Results:

    • The unimodal assumption facilitates efficient computation and more accurate inferences.
    • Using both effect size and standard error accounts for measurement precision variations.
    • The approach provides interval estimates (credible regions) for effect sizes.
    • The local false sign rate is shown to be a superior significance measure compared to local FDR.

    Conclusions:

    • The new Empirical Bayes method offers a robust and accurate approach to large-scale hypothesis testing and FDR estimation.
    • The unimodal assumption and use of dual input data improve statistical inference.
    • The local false sign rate provides a valuable and more generally applicable significance measure.